Specifying a topology by connected subspaces Let $\Gamma_1$ and $\Gamma_2$ be topologies on a set $X$ such that: 
for every subspace $A \subseteq X$, $A$ is connected in $\Gamma_1$ iff it is connected in $\Gamma_2$ 
Are $\Gamma_1$ and $\Gamma_2$ homeomorphic?
 A: No.
Let $X=\{a,b\}$ and let $\Gamma_1$ be the indiscrete topology and $Γ_2$ the Sierpiński topology, where the open sets are $\emptyset, X,$ and $\{a\}$. Then all subsets of $X$ are connected in both, the $Γ_1$ and the $Γ_2$ topology.
A: Stefan Hamcke's example is excellent, but after seeing it, I wondered whether one could construct a Hausdorff counterexample. Indeed, you can. Consider $\mathbb R$ with the standard topology and with the $K$-topology, $\mathbb R_K$. The $K$-topology has as a basis standard open intervals $(a,b)$ as well as sets of the form $(a,b)\setminus K$ where $K=\{\frac{1}{n}\,|\, n\in\mathbb N\}.$ It is easy to see that the subspaces $L=(-\infty, 0]$ and $R=(0,\infty)$ are inherit the same topology from $\mathbb R$ as from $\mathbb R_K$ . Now let $A$ be a subset of $\mathbb R$. If it is disconnected in the standard topology, then it is disconnected in the finer $K$-topology. If it is connected, then it is an interval. If this interval lies in one of the two subspaces $L$ and $R$ then it will be connected since they inherit the standard topology.  Otherwise  $A\cap L$ and $A\cap R$ are both nonempty intervals and are connected in $L$ and $R$ and hence in the $K$-topology. Furthermore $0$ is a limit point of $A\cap R$, so $A$ is the union of two connected sets one of which contains a limit point of the other. Hence $A$ is connected.
A: There are lots of easy counterexamples with totally disconnected spaces.
Example: $X=\mathbb Q$, the set of all rational numbers; $\Gamma_1$ is the usual topologyof $\mathbb Q$; $\Gamma_2$ is the discrete topology.
Another example, this time with two compact metric spaces: $X$ is the Cantor set; $\Gamma_1$ is the usual topology of $X$; $\Gamma_2$ is a topology on $X$ which is homeomorphic to the usual topology of $X\cup\{2\}$.
